With an incomplete solution. October 26, 1999

Rachel has the following utility function
`u(x _{1},x_{2})=½ln(x_{1})+ln(x_{2})`.

**1.1** Find Rachel's demand functions for
`x _{1}` and

**1.2** In the initial state Rachel had an income of `m=100` and
the prices were `p _{1}=p_{2}=2`. Now the price of commodity

Find the change in the quantity of
`x _{1}` demanded,
that occurred due to the price change (find the initial and final bundles),
and decompose it to a change due to the income effect and a change due to
the substitution effect.

Explain the distinction between the two effects with the help of a diagram.

The demand functions are `x _{1}=m/3p_{1}` and

To decompose the change we find the budget line corresponding to the pivotal change.
The prices correspond to the final situation and the income is
`m"=1×100/6+2×200/6=500/6`. The optimal quantity of
`x _{1}` is

**1.3** Sarah has the utility function
`u(x _{1},x_{2})=ln(x_{1})+x_{2}`.
Consider the change in quantity of

In this case the income effect would be zero, as `x _{1}` is a neutral good,
and its level of consumption is independent of the income (in the relevant region of prices
and income).

**1.4** Find the change in Rachel's (consumer's) surplus due to
the price change.

The change in consumers' surplus, measured in units of `x _{2}`
is

2 / |m/p_{2}|----- dp_{1}= [100/6]ln2 |3p_{1}/ 1

Jack consumes bread and wine. The price of bread is `p _{2}` dollars per loaf,
and of wine

**2.1** Draw as diagram of Jack's budget set, his demanded bundle and
the indiference curve that correspondes this bundle.

**2.2** Assume now that the government taxes wine
with `t` dollars per bottle. Draw the new budget line and the new demanded bundle.

**2.3** The government now considers to revise its taxation policy and
impose on Jack a tax of `w` dollars. This amount is fixed and replaces the tax on
wine. Will the government be able to collect more taxes than with the previous taxation policy,
assuming that Jack's utility is identical in both taxation schemes?

See Section 5.6 in the textbook for a solution. In addition, there may be situations that the indeference curves have a kink. In this case it could be that both taxation policies would lead to exactly the same outcome.

A consumer has the utility function

x_{1}if x_{1}<1 U(x_{1},x_{2})= ln[x_{1}]+ln[x_{2}+1]+1 otherwise

Draw a map of the consumer's indiference curves. Draw the income offer curve for the prices
`p _{1}=p_{2}=1`. Draw the Engel curves for both

The indiference curves are vertical parallel lines in the region `x _{1}<1`
and have a regular convex-preference shape otherwise. In this region the indiference curves intersect
with both the horizontal and vertical axes.

The income offer curve goes horizontally from (0,0) until (1,0). From then on it is a straight
line that satisfies the equation `x _{1}=x_{2}+1`.